A Neighbouring Gray Tone Difference Matrix quantifies the difference between a gray value and the average gray value of its neighbours within distance \delta. The sum of absolute differences for gray level i is stored in the matrix. Let \textbf{X}_{gl} be a set of segmented voxels and x_{gl}(j_x,j_y,j_z) \in \textbf{X}_{gl} be the gray level of a voxel at position (j_x,j_y,j_z), then the average gray level of the neighborhood is:
\bar{A}_i &= \bar{A}(j_x, j_y, j_z) \\
&= \frac{1}{W} \sum_{k_x=-\delta}^{\delta}\sum_{k_y=-\delta}^{\delta} \sum_{k_z=-\delta}^{\delta}{x_{gl}(j_x+k_x, j_y+k_y, j_z+k_z)},
where
(k_x,k_y,k_z)\neq(0,0,0)
and
x_{gl}(j_x+k_x, j_y+k_y, j_z+k_z) \in \textbf{X}_{gl}
Here, W is the number of voxels in the neighbourhood that are also in \textbf{X}_{gl}.
As a two dimensional example, let the following matrix \textbf{I} represent a 4x4 image, having 5 discrete grey levels, but no voxels with gray level 4:
\textbf{I} = \begin{bmatrix}
1 & 2 & 5 & 2\\
3 & 5 & 1 & 3\\
1 & 3 & 5 & 5\\
3 & 1 & 1 & 1\end{bmatrix}
The following NGTDM can be obtained:
\begin{array}{cccc}
i & n_i & p_i & s_i\\
\hline
1 & 6 & 0.375 & 13.35\\
2 & 2 & 0.125 & 2.00\\
3 & 4 & 0.25 & 2.63\\
4 & 0 & 0.00 & 0.00\\
5 & 4 & 0.25 & 10.075\end{array}
6 pixels have gray level 1, therefore:
s_1 = |1-10/3| + |1-30/8| + |1-15/5| + |1-13/5| + |1-15/5| + |1-11/3| = 13.35
For gray level 2, there are 2 pixels, therefore:
s_2 = |2-15/5| + |2-9/3| = 2
Similar for gray values 3 and 5:
s_3 = |3-12/5| + |3-18/5| + |3-20/8| + |3-5/3| = 3.03
s_5 = |5-14/5| + |5-18/5| + |5-20/8| + |5-11/5| = 10.075
Let:
n_i be the number of voxels in X_{gl} with gray level i
N_{v,p} be the total number of voxels in X_{gl} and equal to \sum{n_i} (i.e. the number of voxels with a valid region; at least 1 neighbor). N_{v,p} \leq N_p, where N_p is the total number of voxels in the ROI.
p_i be the gray level probability and equal to n_i/N_v
s_i = \sum^{n_i}{|i-\bar{A}_i|} when n_i \neq 0 and s_i = 0 when n_i = 0.
be the sum of absolute differences for gray level i.
N_g be the number of discrete gray levels
N_{g,p} be the number of gray levels where p_i \neq 0
NGTDM_COARSENESS = \frac{1}{\sum^{N_g}_{i=1}{p_{i}s_{i}}}
Assuming p_i and p_j are row indices of the NGTDM matrix,
NGTDM_CONTRAST = \left(\frac{1}{N_{g,p}(N_{g,p}-1)}\sum^{N_g}_{i=1}\sum^{N_g}_{j=1}{p_{i}p_{j}(i-j)^2}\right) \left(\frac{1}{N_{v,p}}\sum^{N_g}_{i=1}{s_i}\right) where p_i \neq 0 and p_j \neq 0
NGTDM_BUSYNESS = \frac{\sum^{N_g}_{i = 1}{p_{i}s_{i}}}{\sum^{N_g}_{i = 1}\sum^{N_g}_{j = 1}{|ip_i - jp_j|}} where p_i \neq 0, p_j \neq 0
NGTDM_COMPLEXITY = \frac{1}{N_{v,p}}\sum^{N_g}_{i = 1}\sum^{N_g}_{j = 1}{|i - j| \frac{p_{i}s_{i} + p_{j}s_{j}}{p_i + p_j}} where p_i \neq 0, p_j \neq 0
NGTDM_STRENGTH = \frac{\sum^{N_g}_{i = 1}\sum^{N_g}_{j = 1}{(p_i + p_j)(i-j)^2}}{\sum^{N_g}_{i = 1}{s_i}} where p_i \neq 0, p_j \neq 0
Amadasun M, King R; Textural features corresponding to textural properties; Systems, Man and Cybernetics, IEEE Transactions on 19:1264-1274 (1989). doi: 10.1109/21.44046